A121013 -id:A121013 - cp's OEIS frontend

A100402 Digital root of 4^n.

1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7, 1, 4, 7

Offset: 0

Cino Hilliard, Dec 31 2004

Equals A141725 mod 9. - Paul Curtz, Sep 15 2008
Sequence is the digital root of A016777. - Odimar Fabeny, Sep 13 2010
Digital root of the powers of any number congruent to 4 mod 9. - Alonso del Arte, Jan 26 2014
Period 3: repeat [1, 4, 7]. - Wesley Ivan Hurt, Aug 26 2014
From Timothy L. Tiffin, Dec 02 2023: (Start)
The period 3 digits of this sequence are the same as those of A070403 (digital root of 7^n) but the order is different: [1, 4, 7] vs. [1, 7, 4].
The digits in this sequence appear in the decimal expansions of the following rational numbers: 49/333, 490/333, 4900/333, .... (End)

			4^2 = 16, digitalroot(16) = 7, the third entry.

Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. Digital roots of powers of c mod 9: c = 2, A153130; c = 5, A070366; c = 7, A070403; c = 8, A010689.

Cf. A000302, A001022, A009966, A009975, A009984, A010872, A010888, A070403, A087752, A121013, A141725, A016777.

Table[PowerMod[4, n, 9], {n, 0, 100}] (* Timothy L. Tiffin, Dec 03 2023 *)
StringRepeat["1, 4, 7, ", 100] (* Timothy L. Tiffin, Dec 03 2023 *)

a(n)=[1,4,7][1+n%3]; \\ Joerg Arndt, Aug 26 2014

[power_mod(4, n, 9) for n in range(0, 105)] # Zerinvary Lajos, Nov 25 2009

a(n) = 4^n mod 9. - Zerinvary Lajos, Nov 25 2009
From R. J. Mathar, Apr 13 2010: (Start)
a(n) = a(n-3) for n>2.
G.f.: (1+4*x+7*x^2)/ ((1-x)*(1+x+x^2)). (End)
a(n) = A010888(A000302(n)). - Michel Marcus, Aug 25 2014
a(n) = 3*A010872(n) + 1. - Robert Israel, Aug 25 2014
a(n) = 4 - 3*cos(2*n*Pi/3) - sqrt(3)*sin(2*n*Pi/3). - Wesley Ivan Hurt, Jun 30 2016
a(n) = A153130(2n). - Timothy L. Tiffin, Dec 01 2023
a(n) = A010888(A001022(n)) = A010888(A009966(n)) = A010888(A009975(n)) = A010888(A009984(n)) = A010888(A087752(n)) = A010888(A121013(n)). - Timothy L. Tiffin, Dec 02 2023
a(n) = A010888(4*a(n-1)). - Stefano Spezia, Mar 20 2025

A121012 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(11^2) = 1/121.

Original entry on oeis.org

1, 120, 14522, 1757157, 212616011, 25726537289, 282991910191, 34242021133072, 4143284557101842, 501337431409322440, 667280121205808184436, 80740894665902790257970, 9769648254574237621422382

Offset: 0

Wolfdieter Lang, Aug 16 2006

Denominators are given under A121013.
This is the second member (p=2) of the fourth (normalized) p-family of partial sums of normalized scaled Catalan series CsnIV(p):=sum(((-1)^k)*C(k)/L(2*p+1)^(2*k),k=0..infinity) with limit L(2*p+1)*(-F(2*p+2) + F(2*p+1)*phi) = L(2*p+1)/phi^(2*p+1), with C(n)=A000108(n) (Catalan), F(n)= A000045(n) (Fibonacci), L(n) = A000032(n) (Lucas) and phi:=(1+sqrt(5))/2 (golden section).
The partial sums of the above mentioned fourth p-family are rIV(p;n):=sum(((-1)^k)*C(k)/L(2*p+1)^(2*k),k=0..n), n>=0, for p=1,...
For more details on this p-family and the other three ones see the W. Lang link under A120996.

			Rationals r(n): [1, 120/121, 14522/14641, 1757157/1771561,
212616011/214358881, 25726537289/25937424601,...].

W. Lang: Rationals r(n), limit.

The first member is A120794/A120785. The third member is A121498/A121499.

The limit lim_{n->infinity} (r(n) := rIV(2;n)) = 11*(-8 + 5*phi) = 11/phi^5 = 0.9918693812443 (maple10, 10 digits).

a(n)=numerator(r(n)) with r(n) := rIV(p=2,n) = sum(((-1)^k)*C(k)/L(2*2+1)^(2*k), k=0..n), with L(5)=11 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

A120794 Numerators of partial sums of Catalan numbers scaled by powers of -1/16.

Original entry on oeis.org

1, 15, 121, 3867, 30943, 495067, 3960569, 253475987, 2027808611, 32444935345, 259559486959, 8305903553295, 66447228478363, 1063155655468083, 8505245244078969, 1088671391232413187

Offset: 0

Wolfdieter Lang, Jul 20 2006

From the expansion of sqrt(1+1/4) = 1+(1/8)*Sum_{k>=0} C(k)/(-16)^k one has, with the partial sums r(n) are defined below, r := lim_{n->oo} r(n) = 4*(sqrt(5)-2) = 4*(2*phi-3) = 0.944271909...
Denominators coincide with the listed numbers of A120785 but may differ for higher n values.
This is the first member (p=1) of the fourth family of scaled Catalan sums with limits in Q(sqrt(5)). See the W. Lang link under A120996.

			Rationals r(n): [1, 15/16, 121/128, 3867/4096, 30943/32768, 495067/524288, 3960569/4194304,...].

Wolfdieter Lang, Rationals r(n) and limit.

The second member (p=2) of this p-family is A121012/A121013.

a(n)=numerator(r(n)), with the rationals r(n):=Sum_{k=0..n} ((-1)^k)*C(k)/16^k with C(k):=A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.

A121498 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(29^2) = 1/841.

Original entry on oeis.org

1, 840, 706442, 594117717, 499653000011, 420208173009209, 353395073500744901, 297205256814126461312, 249949620980680353964822, 210207631244752177684410440, 176784617876836581432589196836

Offset: 0

Wolfdieter Lang, Aug 16 2006

Denominators are given under A121499.
This is the third member (p=3) of the fourth (normalized) p-family of partial sums of normalized scaled Catalan series CsnIV(p):=sum(((-1)^k)*C(k)/L(2*p+1)^(2*k),k=0..infinity) with limit L(2*p+1)*(-F(2*p+2) + F(2*p+1)*phi) = L(2*p+1)/phi^(2*p+1), with C(n)=A000108(n) (Catalan), F(n)= A000045(n) (Fibonacci), L(n) = A000032(n) (Lucas) and phi:=(1+sqrt(5))/2 (golden section).
The partial sums of the above mentioned fourth p-family are rIV(p;n):=sum(((-1)^k)*C(k)/L(2*p+1)^(2*k),k=0..n), n>=0, for p=1,...
For more details on this p-family and the other three ones see the W. Lang link under A120996.

			Rationals r(n): [1, 840/841, 706442/707281, 594117717/594823321,
499653000011/500246412961, 420208173009209/420707233300201,...].

W. Lang: Rationals r(n), limit.

The second member (p=2) of this p-family is A121012/A121013.

The limit lim_{n->infinity}(r(n) := rIV(2;n)) = 29*(-21 + 13*phi) = 29/phi^7 = 0.998813758709 (maple10, 10 digits).

a(n)=numerator(r(n)) with r(n) := rIV(p=3,n) = sum(((-1)^k)*C(k)/L(2*3+1)^(2*k),k=0..n), with L(7)=29 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

A121499 Denominators of partial alternating sums of Catalan numbers scaled by powers of 1/(29^2) = 1/841.

Original entry on oeis.org

1, 841, 707281, 594823321, 500246412961, 420707233300201, 353814783205469041, 297558232675799463481, 250246473680347348787521, 210457284365172120330305161, 176994576151109753197786640401

Offset: 0

Wolfdieter Lang, Aug 16 2006

Numerators are given under A121498.
This is the third member (p=3) of the fourth (normalized) p-family of partial sums of normalized scaled Catalan series CsnIV(p):=sum(((-1)^k)*C(k)/L(2*p+1)^(2*k),k=0..infinity) with limit L(2*p+1)*(-F(2*p+2) + F(2*p+1)*phi) = L(2*p+1)/phi^(2*p+1), with C(n)=A000108(n) (Catalan), F(n)= A000045(n) (Fibonacci), L(n) = A000032(n) (Lucas) and phi:=(1+sqrt(5))/2 (golden section).
The partial sums of the above mentioned fourth p-family are rIV(p;n):=sum(((-1)^k)*C(k)/L(2*p+1)^(2*k),k=0..n), n>=0, for p=1,...
For more details on this p-family and the other three ones see the W. Lang links under A120996 and A121498.

			Rationals r(n): [1, 840/841, 706442/707281, 594117717/594823321,
499653000011/500246412961, 420208173009209/420707233300201,...].

Tanya Khovanova, Non Recursions

The second member (p=2) of this p-family is A121012/A121013.

a(n)=denominator(r(n)) with r(n) := rIV(p=3,n) = sum(((-1)^k)*C(k)/L(2*3+1)^(2*k),k=0..n), with L(7)=29 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

cp's OEIS Frontend

A100402 Digital root of 4^n.

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A121012 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(11^2) = 1/121.

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A120794 Numerators of partial sums of Catalan numbers scaled by powers of -1/16.

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A121498 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(29^2) = 1/841.

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A121499 Denominators of partial alternating sums of Catalan numbers scaled by powers of 1/(29^2) = 1/841.

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