A153286 -id:A153286 - cp's OEIS frontend

A153284 a(n) = n + Sum_{j=1..n-1} (-1)^j * a(j) for n >= 2, a(1) = 1.

1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1

Offset: 1

Walter Carlini, Dec 23 2008

Row sums of triangle A153860. - Gary W. Adamson, Jan 03 2009

1 followed by interleaving of A000012 and A010701. - Klaus Brockhaus, Jan 04 2009

			a(1)=1, a(2)=2-a(1)=2-1=1, a(3)=3+a(2)-a(1)=3+1-1=3, a(4)=4-a(3)+a(2)-a(1)=4-3+1-1=1, a(5)=5+1-3+1-1=3, a(6)=6-3+1-3+1-1=1, a(7)=7+1-3+1-3+1-1, etc.

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

Equals A010684 with the addition of the leading term of 1
The first sequence of a family that includes A153285 and A153286
Cf. A153860.
Cf. A000012 (all 1's sequence), A010701 (all 3's sequence). - Klaus Brockhaus, Jan 04 2009

S:=[ 1 ]; for n in [2..105] do Append(~S, n + &+[ (-1)^j*S[j]: j in [1..n-1] ]); end for; S; // Klaus Brockhaus, Jan 04 2009

a(n)=1 if n is 1 or even; a(n)=3 if n is odd other than 1.

G.f.: x*(1 + x + 2*x^2)/((1+x)*(1-x)). - Klaus Brockhaus, Jan 04 2009 and Oct 15 2009

A154105 a(n) = 12n^2 + 18n + 7.

Original entry on oeis.org

7, 37, 91, 169, 271, 397, 547, 721, 919, 1141, 1387, 1657, 1951, 2269, 2611, 2977, 3367, 3781, 4219, 4681, 5167, 5677, 6211, 6769, 7351, 7957, 8587, 9241, 9919, 10621, 11347, 12097, 12871, 13669, 14491, 15337, 16207, 17101, 18019, 18961, 19927, 20917, 21931

Offset: 0

Klaus Brockhaus, Jan 04 2009

a(n) is the number of partitions with three integral dissimilar components of the number 12(n+1), e.g for n=0, 12 may be partitioned in the 7 ways (1,2,9), (1,3,8), (1,4,7), (1,5,6), (2,3,7), (2,4,6) and (3,4,5). - Ian Duff, Jan 31 2010

Sequence found by reading the line from 7, in the direction 7, 37, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, May 08 2018

			a(2) = 12*2^2 + 18*2 + 7 = 91 = 6*14 + 7 = 6*A014106(2) + 7.
a(3) = a(2) + 24*3 + 6 = 91 + 72 + 6 = 169.
a(-4) = 12*4^2 - 18*4 + 7 = 127 = 2*64 - 1 = 2*A085473(3) - 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
John Elias, Animated Illustration: Starburst Hexagrams
Leo Tavares, Illustration: Hexagonal Halos
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001082, A014106, A152746, A153286, A085473.

Cf. A003215, A000290, A000217.

Magma
```
[ 12*n^2+18*n+7: n in [0..40] ];
```

Table[12*n^2 + 18*n + 7, {n, 0, 42}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)
LinearRecurrence[{3,-3,1}, {7,37,91}, 25] (* G. C. Greubel, Sep 02 2016 *)

a(n)=12*n^2+18*n+7 \\ Charles R Greathouse IV, Sep 02 2016

G.f.: (7 + 16*x + x^2)/(1-x)^3.
a(n) = 6*A014106(n) + 7.
a(0) = 7; for n > 0, a(n) = a(n-1) + 24*n + 6.
a(-n-1) = 2*A085473(n) - 1. - Bruno Berselli, Sep 05 2011
E.g.f.: (7 + 30*x + 12*x^2)*exp(x). - G. C. Greubel, Sep 02 2016
a(n) = 1 + A152746(n+1). - Omar E. Pol, May 08 2018
a(n) = A003215(n) + 6*A000290(n+1) + 6*A000217(n). - Leo Tavares, Sep 12 2022

A154106 a(n) = 12n^2 + 22n + 11.

Original entry on oeis.org

11, 45, 103, 185, 291, 421, 575, 753, 955, 1181, 1431, 1705, 2003, 2325, 2671, 3041, 3435, 3853, 4295, 4761, 5251, 5765, 6303, 6865, 7451, 8061, 8695, 9353, 10035, 10741, 11471, 12225, 13003, 13805, 14631, 15481, 16355, 17253, 18175, 19121

Offset: 0

Klaus Brockhaus, Jan 04 2009

Sequence found by reading the line from 11, in the direction 11, 45,..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012

			a(3) = 12*3^2 + 22*3 + 11 = 185 = 2*3*29 + 11 = 2*3*A016969(4) + 11.
a(4) = a(3) +24*4 +10 = 185 +96 +10 = 291.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016969 (6n+5), A153286, A185918, A194454.

Magma
```
[ 12*n^2+22*n+11: n in [0..39] ];
```

Table[12n^2+22n+11,{n,0,50}]  (* Harvey P. Dale, Mar 16 2011 *)
LinearRecurrence[{3,-3,1},{11,45,103}, 25] (* G. C. Greubel, Sep 02 2016 *)

a(n)=12*n^2+22*n+11 \\ Charles R Greathouse IV, Oct 16 2015

G.f.: (1 +x)*(11 +x)/(1-x)^3.
a(n) = 2*n*A016969(n+1) + 11.
a(0) = 11; for n > 0, a(n) = a(n-1) + 24*n + 10.
a(n) = 2 + A185918(n+1). - Omar E. Pol, Jul 18 2012
E.g.f.: (11 + 34*x + 12*x^2)*exp(x). - G. C. Greubel, Sep 02 2016

A153285 a(1)=1; for n > 1, a(n) = n^2 + Sum_{j=1..n-1} (-1)^j*a(j).

Original entry on oeis.org

1, 3, 11, 7, 23, 11, 35, 15, 47, 19, 59, 23, 71, 27, 83, 31, 95, 35, 107, 39, 119, 43, 131, 47, 143, 51, 155, 55, 167, 59, 179, 63, 191, 67, 203, 71, 215, 75, 227, 79, 239, 83, 251, 87, 263, 91, 275, 95, 287, 99, 299, 103, 311, 107, 323, 111, 335, 115, 347, 119, 359

Offset: 1

Walter Carlini, Dec 23 2008

1 followed by interleaving of A004767 and A017653. - Klaus Brockhaus, Jan 04 2009

			a(1) = 1;
a(2) = 2^2 - a(1) = 4 - 1 = 3;
a(3) = 3^2 + a(2) - a(1) = 9 + 3 - 1 = 11;
a(4) = 4^2 - 11 + 3 - 1 = 7;
a(5) = 25 + 7 - 11 + 3 - 1 = 23;
a(6) = 36 - 23 + 7 - 11 + 3 - 1 = 11; etc.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

The second of a family of sequences that includes A153284 and A153286

Cf. A004767 (4n+3), A017653 (12n+11). - Klaus Brockhaus, Jan 04 2009

S:=[ 1 ]; for n in [2..61] do Append(~S, n^2 + &+[ (-1)^j*S[j]: j in [1..n-1] ]); end for; S; // Klaus Brockhaus, Jan 04 2009

(define (A153285 n) (cond ((= 1 n) n) ((even? n) (+ n n -1)) (else (+ (* 6 n) -7)))) ;; Antti Karttunen, Aug 10 2017

a(n) = 2n-1 if n is 1 or an even number;
a(n) = 6n-7 if n is an odd number other than 1.
G.f.: x*(1 + 3*x + 9*x^2 + x^3 + 2*x^4)/((1+x)^2*(1-x)^2). - Klaus Brockhaus, Oct 15 2009
a(n) = 4*(n-1) - (2*n-3)*(-1)^n for n>1, a(1)=1. - Bruno Berselli, Sep 14 2011

Extended beyond a(30) by Klaus Brockhaus, Jan 04 2009

cp's OEIS Frontend

A153284 a(n) = n + Sum_{j=1..n-1} (-1)^j * a(j) for n >= 2, a(1) = 1.

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A154105 a(n) = 12*n^2 + 18*n + 7.

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A154106 a(n) = 12*n^2 + 22*n + 11.

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A153285 a(1)=1; for n > 1, a(n) = n^2 + Sum_{j=1..n-1} (-1)^j*a(j).

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A154105 a(n) = 12n^2 + 18n + 7.

A154106 a(n) = 12n^2 + 22n + 11.