A371707 Constant r > 0 satisfying: Sum_{n>=0} (r^n + 2*Pi*i)^n/n! = C + i*S such that C^2 + S^2 = 1.

2, 7, 3, 4, 3, 9, 0, 0, 1, 9, 0, 8, 5, 6, 8, 3, 8, 5, 5, 3, 8, 7, 9, 1, 7, 5, 8, 0, 0, 4, 6, 9, 8, 1, 5, 0, 2, 4, 0, 1, 7, 4, 5, 5, 6, 0, 1, 9, 5, 3, 7, 4, 0, 3, 7, 9, 5, 7, 8, 7, 7, 4, 6, 4, 5, 0, 9, 3, 5, 0, 8, 6, 8, 8, 7, 8, 4, 2, 8, 6, 6, 5, 9, 7, 5, 4, 3, 3, 8, 7, 4, 2, 2, 9, 6, 2, 1, 9, 5, 2

Offset

0,1

Comments

Related identity: Sum_{n>=0} (x^n + y)^n/n! = Sum_{n>=0} exp(y*x^n)*x^(n^2)/n!. Here, x = r and y = 2*Pi*i.

What are the roots of Norm( Sum_{n>=0} (x^n + 2*Pi*i)^n/n! ) = 1? The real roots include x = 0 and x = r (this constant).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Formula

Constant r and related values C and S satisfy the following formulas.

(1) Sum_{n>=0} (r^n + 2*Pi*i)^n/n! = C + i*S such that C^2 + S^2 = 1.

(2) C = Sum_{n>=0} cos(2*Pi*r^n) * r^(n^2) / n!.

(3) S = Sum_{n>=0} sin(2*Pi*r^n) * r^(n^2) / n!.

Example

The initial 500 digits of this constant r are
r = 0.27343900190856838553879175800469815024017455601953\
74037957877464509350868878428665975433874229621952\
21271807208862504474781327669150216691806622917186\
30052292342530146845288659570856888661537928135397\
91914154858221560663972999347727219299210079054658\
20785838554943078876634169703813817526574697076018\
43103025671330263969269247113168608393647224573552\
82695245129846145197371729802801821910764770241403\
85315562772171090016733480930506290614196661276630\
35680469795753191100711562687066719873558759501438...
Given Sum_{n>=0} (r^n + 2*Pi*i)^n / n! = C + i*S
then C = Sum_{n>=0} cos(2*Pi*r^n) * r^(n^2) / n!, where
C = 0.96236940120128609855708390989630224707797733780139\
33346689286186097367092064030604732865267035268595\
44279783779811281344593178122348416729686502694192\
27215955652725928674242226419071059523037649451781\
36060669147586159699815697962817267659814744582224\
93126268783872251860132042094952557434607056861286\
20902477149931860926346847824008347947488598827305\
47837372109484356517193566333052743194953698066525\
72228584587713864226102674129509160583381421007047\
75118828482389128699072732009353421657729660481717...
and S = Sum_{n>=0} sin(2*Pi*r^n) * r^(n^2) / n!, where
S = 0.27174461472396842102515050866607715426951746748919\
04159412993022348271493896385066506863889535797824\
35085649751784233166430963459007191963331589808443\
52259856849111637575812332490848107413710402589323\
75221334357855133874979455560441735994213395179878\
38917993730963815574520261440791182088848636006332\
68221934823032560291871222621378256174374612116671\
09358271083370500808439006024716239994653435216572\
21204963868973568338610259219318795040671357965613\
68248089245008828798740589773672045329008665505374...
such that C^2 + S^2 = 1.

Crossrefs

Cf. A326600.

Sequence in context: A088538 A210516 A257102 * A226626 A249778 A350505

Adjacent sequences: A371704 A371705 A371706 * A371708 A371709 A371710

Keyword

nonn,cons

Author

Paul D. Hanna, Apr 09 2024

Status

approved