A050807 -id:A050807 - cp's OEIS frontend

A050819 Increasing odd numbers seen in decimal expansion of Pi (disregarding the decimal period) contiguous, smallest and distinct.

3, 141, 59265, 358979, 3238462643, 3832795028841, 97169399375105, 820974944592307, 81640628620899862803, 48253421170679821480865, 132823066470938446095505, 82231725359408128481117450284102701, 9385211055596446229489549303819644288109

Offset: 1

Patrick De Geest, Oct 15 1999

Leading zero not allowed thus forcing continuation of previous term.

Carlos Rivera, Problem 18. Pi as a concatenation of the smallest contiguous different primes, The Prime Puzzles & Problems Connection.

Cf. A000796, A016062, A037244, A047777, A050817, A016062, A050818, A050807.

a(12) corrected and a(13) from Sean A. Irvine, Aug 19 2021

A050818 Even numbers seen in decimal expansion of Pi (disregarding the decimal period) contiguous, smallest and distinct.

Original entry on oeis.org

314, 1592, 6, 5358, 97932, 38, 4, 62, 64, 338, 32, 7950, 2, 8, 84, 19716, 939937510, 58, 20, 974, 94, 4592, 30, 78, 16, 40, 628, 620, 8998, 6280, 34, 82, 534, 21170, 6798, 214, 80, 86, 5132, 8230, 66, 470, 938, 44, 60, 9550, 582, 23172, 53594, 0, 812, 848

Offset: 1

Patrick De Geest, Oct 15 1999

Leading zero not allowed thus forcing continuation of previous term.

C. Rivera, Problem 18, The Prime Puzzles & Problems Connection.

Cf. A047777, A050817, A016062, A050819, A050807.

A050817 Odd numbers seen in decimal expansion of Pi (disregarding the decimal period) contiguous, smallest and distinct.

Original entry on oeis.org

3, 1, 41, 5, 9, 265, 35, 89, 7, 93, 23, 8462643, 383, 27, 95028841, 97, 169, 39, 937, 5105, 8209, 749, 445, 92307, 81, 64062862089, 9862803, 4825, 3421, 17067, 9821, 480865, 13, 28230664709, 3844609, 5505, 8223, 17, 25, 359, 4081, 28481, 11

Offset: 1

Patrick De Geest, Oct 15 1999

Leading zero not allowed thus forcing continuation of previous term.

C. Rivera, Problem 18, The Prime Puzzles & Problems Connection.

Cf. A047777, A050818, A016062, A050819, A050807.

A131751 Numbers that are both centered triangular and centered pentagonal.

Original entry on oeis.org

1, 31, 1891, 117181, 7263301, 450207451, 27905598631, 1729696907641, 107213302675081, 6645495068947351, 411913480972060651, 25531990325198812981, 1582571486681354344141, 98093900183918770523731, 6080239239916282418127151, 376876738974625591153359601

Offset: 1

Richard Choulet, Sep 20 2007

We solve 0.5*(3*p^2+3*p+2)=0.5*(5*r^2+5*r+2), i.e., 3*(2*p+1)^2=5*(2*r+1)^2-2.

The Diophantine equation 3*X^2=5*Y^2-2is such that : X is given by A057080 which satisfies the new formula a(n+1)=4*a(n)+(15*a(n)^2+10)^0.5, Y is given by A070997 which satisfies the new formula a(n+1)=4*a(n)+(15*a(n)^2-6)^0.5 while r is given by the sequence 0,3,27,216,1704,... which satisfies a(n+2)=8*a(n+1)-a(n)+3 and a(n+1)=4*a(n)+1.5+0.5*(60*a(n)^2+60*a(n)+9)^0.5, p is given by the sequence 0,4,35,279,2200,... which satisfies a(n+2)=8*a(n+1)-a(n)+3 and a(n+1)=4*a(n)+1.5+0.5*sqrt(60*a(n)^2+60*a(n)+25).

Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.
Index entries for linear recurrences with constant coefficients, signature (63,-63,1).

Cf. A050807 A070997.

A131751 := proc(n) coeftayl(x*(1-32*x+x^2)/(1-x)/(1-62*x+x^2),x=0,n) ; end: seq(A131751(n),n=1..20) ; # R. J. Mathar, Oct 24 2007

LinearRecurrence[{63,-63,1},{1,31,1891},20] (* Harvey P. Dale, Oct 01 2017 *)

a(n+2) = 62*a(n+1) - a(n) - 30, a(n+1) = 31*a(n) - 15 + sqrt(960*a(n)^2 - 960*a(n)+225).
G.f.: f(z) = a(1)*z+a(2)*z^2+... = z*(1-32*z+z^2)/((1-z)*(1-62*z+z^2)).
A005891 INTERSECT A005448. - R. J. Mathar, Oct 24 2007

Corrected and extended by R. J. Mathar, Oct 24 2007

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A050819 Increasing odd numbers seen in decimal expansion of Pi (disregarding the decimal period) contiguous, smallest and distinct.

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A050818 Even numbers seen in decimal expansion of Pi (disregarding the decimal period) contiguous, smallest and distinct.

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A050817 Odd numbers seen in decimal expansion of Pi (disregarding the decimal period) contiguous, smallest and distinct.

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A131751 Numbers that are both centered triangular and centered pentagonal.

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